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Mean

In statistics, the mean is one of the measures of central tendency, apart from the mode and median. Mean is nothing but the average of the given set of values. It denotes the equal distribution of values for a given data set. The mean, median and mode are the three commonly used measures of central tendency. To calculate the mean, we need to add the total values given in a datasheet and divide the sum by the total number of values. The Median is the middle value of a given data when all the values are arranged in ascending order. Whereas mode is the number in the list, which is repeated a maximum number of times. 

Learn: Central tendency

In this article, you will learn the definition of mean, the formula for finding the mean for ungrouped and grouped data, along with the applications and solved examples.

Definition of Mean in Statistics

Mean is the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers. 

Mean = (Sum of all the observations/Total number of observations)

Example:

What is the mean of 2, 4, 6, 8 and 10?

Solution:

First, add all the numbers.

2 + 4 + 6 + 8 + 10 = 30

Now divide by 5 (total number of observations).

Mean = 30/5 = 6

In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x) and then adding all these products together.

Mean Symbol (X Bar)

The symbol of mean is usually given by the symbol ‘x̄’. The bar above the letter x, represents the mean of x number of values. 

X̄ = (Sum of values ÷ Number of values)

X̄ = (x1 + x2 + x3 +….+xn)/n

Mean Formula

The basic formula to calculate the mean is calculated based on the given data set. Each term in the data set is considered while evaluating the mean. The general formula for mean is given by the ratio of the sum of all the terms and the total number of terms. Hence, we can say;

Mean = Sum of the Given Data/Total number of Data

To calculate the arithmetic mean of a set of data we must first add up (sum) all of the data values (x) and then divide the result by the number of values (n). Since ∑ is the symbol used to indicate that values are to be summed (see Sigma Notation) we obtain the following formula for the mean (x̄):

x̄=∑ x/n

How to Find Mean?

As we know, data can be grouped data or ungrouped data so to find the mean of given data we need to check whether the given data is ungrouped. The formulas to find the mean for ungrouped data and grouped data are different. In this section, you will learn the method of finding the mean for both of these instances.

Mean for Ungrouped Data

The example given below will help you in understanding how to find the mean of ungrouped data.

Example:

In a class there are 20 students and they have secured a percentage of 88, 82, 88, 85, 84, 80, 81, 82, 83, 85, 84, 74, 75, 76, 89, 90, 89, 80, 82, and 83.

Find the mean percentage obtained by the class.

Solution:

Mean = Total of percentage obtained by 20 students in class/Total number of students

= [88 + 82 + 88 + 85 + 84 + 80 + 81 + 82 + 83 + 85 + 84 + 74 + 75 + 76 + 89 + 90 + 89 + 80 + 82 + 83]/20

= 1660/20

= 83

Hence, the mean percentage of each student in the class is 83%.

Mean for Grouped Data

For grouped data, we can find the mean using either of the following formulas.

Direct method:

Mean

\(\begin{array}{l}\overline{x}=\frac{\sum_{i=1}^{n}f_ix_i}{\sum_{i=1}^{n}f_i}\end{array} \)

Assumed mean method:

Mean

\(\begin{array}{l}(\overline{x})=a+\frac{\sum f_id_i}{\sum f_i}\end{array} \)

Step-deviation method:

Mean

\(\begin{array}{l}(\overline{x})=a+h\frac{\sum f_iu_i}{\sum f_i}\end{array} \)

Go through the example given below to understand how to calculate the mean for grouped data.

Example:

Find the mean for the following distribution.

xi 11 14 17 20
fi 3 6 8 7

Solution:

For the given data, we can find the mean using the direct method.

xi fi fixi
11 3 33
14 6 84
17 8 136
20 7 140
∑fi = 24 ∑fi x= 393

Mean = ∑fixi/∑fi = 393/24 = 16.4

Mean of Negative Numbers

We have seen examples of finding the mean of positive numbers till now. But what if the numbers in the observation list include negative numbers. Let us understand with an instance,

Example:

Find the mean of 9, 6, -3, 2, -7, 1.

Solution:

Add all the numbers first:

Total: 9+6+(-3)+2+(-7)+1 = 9+6-3+2-7+1 = 8

Now divide the total from 6, to get the mean.

Mean = 8/6 = 1.33

Types of Mean

There are majorly three different types of mean value that you will be studying in statistics.

  1. Arithmetic Mean
  2. Geometric Mean
  3. Harmonic Mean

Arithmetic Mean

When you add up all the values and divide by the number of values it is called Arithmetic Mean. To calculate, just add up all the given numbers then divide by how many numbers are given.

Example: What is the mean of 3, 5, 9, 5, 7, 2?

Now add up all the given numbers:

3 + 5 + 9 + 5 + 7 + 2 = 31

Now divide by how many numbers are provided in the sequence:

316= 5.16

5.16 is the answer.

Geometric Mean

The geometric mean of two numbers x and y is xy. If you have three numbers x, y, and z, their geometric mean is 3xyz.

\(\begin{array}{l}\large Geometric\;Mean=\sqrt[n]{x_{1}x_{2}x_{3}…..x_{n}}\end{array} \)

Example: Find the geometric mean of 4 and 3 ?

Geometric Mean =

\(\begin{array}{l} \sqrt{4 \times 3} = 2 \sqrt{3} = 3.46\end{array} \)

Harmonic Mean

The harmonic mean is used to average ratios. For two numbers x and y, the harmonic mean is 2xy(x+y). For, three numbers x, y, and z, the harmonic mean is 3xyz(xy+xz+yz)

\(\begin{array}{l}\large Harmonic\;Mean (H) = \frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+……\frac{1}{x_{n}}}\end{array} \)

Root Mean Square (Quadratic)

The root mean square is used in many engineering and statistical applications, especially when there are data points that can be negative.

\(\begin{array}{l}\large X_{rms}=\sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}….x_{n}^{2}}{n}}\end{array} \)

Contraharmonic Mean

The contraharmonic mean of x and y is (x2 + y2)/(x + y). For n values,

\(\begin{array}{l}\large \frac{(x_{1}^{2}+x_{2}^{2}+….+x_{n}^{2})}{(x_{1}+x_{2}+…..x_{n})}\end{array} \)

Real-life Applications of Mean

In the real world, when there is huge data available, we use statistics to deal with it. Suppose, in a data table, the price values of 10 clothing materials are mentioned. If we have to find the mean of the prices, then add the prices of each clothing material and divide the total sum by 10. It will result in an average value. Another example is that if we have to find the average age of students of a class, we have to add the age of individual students present in the class and then divide the sum by the total number of students present in the class.

Practice Problems

Q.1: Find the mean of 5,10,15,20,25.

Q.2: Find the mean of the given data set: 10,20,30,40,50,60,70,80,90.

Q.3: Find the mean of the first 10 even numbers.

Q.4: Find the mean of the first 10 odd numbers.

Frequently Asked Questions – FAQs

What is mean in statistics?

In statistics, Mean is the ratio of sum of all the observations and total number of observations in a data set. For example, mean of 2, 6, 4, 5, 8 is:
Mean = (2 + 6 + 4 + 5 + 8) / 5 = 25/5 = 5

How is mean represented?

Mean is usually represented by x-bar or x̄.
X̄ = (Sum of values ÷ Number of values in data set)

What is median in Maths?

Median is the central value of the data set when they are arranged in an order.
For example, the median of 3, 7, 1, 4, 8, 10, 2.
Arrange the data set in ascending order: 1,2,3,4,7,8,10
Median = middle value = 4

What are the types of Mean?

In statistics we learn basically, three types of mean, they are:
Arithmetic Mean, Geometric Mean and Harmonic Mean

What is the mean of the first 10 natural numbers?

The first 10 natural numbers are: 1,2,3,4,5,6,7,8,9,10
Sum of first 10 natural numbers = 1+2+3+4+5+6+7+8+9+10 = 55
Mean = 55/10 = 5.5

What is the relationship between mean, median and mode?

The relationship between mean, median and mode is given by:
3 Median = Mode + 2 Mean.

What is the mean of the first 5 even natural numbers?

As we know, the first 5 even natural numbers are 2, 4, 6, 8, and 10.
Hence, Mean = (2 + 4 + 6 + 8 + 10)/5
Mean = 6
Thus, the mean of the first 5 even natural numbers is 6.

Find the mean of the first 5 composite numbers?

The first 5 composite numbers are 4, 6, 8, 9 and 10.
Thus, Mean = (4 + 6 + 8 + 9 + 10)/5
Mean = 37/5 = 7.4
Hence, the mean of the first 5 composite numbers is 7.4.

Test your knowledge on Mean

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